Question

How do you find the LCM of \[\left( x-1 \right)\left( x+2 \right)\] and \[\left( x-1 \right)\left( x+3 \right)\]?

Answer 1

Hint: The Highest common factor of two numbers is the largest number that can divide both the given numbers. The Lowest common multiple of two numbers is the smallest number which can be divided by both of the given numbers. If a, and b are the two numbers, and H and L are their Highest common factor and Lowest common multiple respectively, then we can use the formula \[ab=HL\], which means that the product of two numbers is the same as the product of their highest common factor and lowest common multiple.

Complete step by step solution:
We are given two expressions as \[\left( x-1 \right)\left( x+2 \right)\] and \[\left( x-1 \right)\left( x+3 \right)\]. We need to find their LCM, first let’s find their HCF. To do this we have to express the given expressions in factored form.
As we can see that both the expressions are already given in their factored form, and the only common factor both of them have is \[\left( x-1 \right)\]. Thus, the highest common factor of the expressions is \[\left( x-1 \right)\].
We can find the Lowest common multiple or LCM by using the property that the product of two numbers is the same as the product of their highest common factor and lowest common multiple. We get,
\[\Rightarrow \left( x-1 \right)\left( x+2 \right)\left( x-1 \right)\left( x+3 \right)=\left( x-1 \right)\times LCM\]
Solving the above equation, we get
\[\Rightarrow LCM=\left( x-1 \right)\left( x+2 \right)\left( x+3 \right)\]

Hence, the LCM of the two expressions is \[\left( x-1 \right)\left( x+2 \right)\left( x+3 \right)\].

Note: To solve these types of problems, it is easier to find the highest common factor first and then the lowest common multiple using the formula. Finding the HCF of a number is easier than finding the LCM of any two given numbers.